Lesson 1: Noetherianity versus coherence.
Finitely generated abelian groups, finitely generated modules over a PID. (structure theorem, classical and constructive, constructive noetheriannity)
http://hlombardi.free.fr/publis/LectSlides1.pdf
Lesson 2: (elimination of primes)
Krull dimension (classical and constructive, constructive treatment of some "great" theorems)
http://hlombardi.free.fr/publis/LectSlides2.pdf
Lesson 3: (elimination of minimal primes)
Seminormal rings. How to eliminate a minimal prime in a classical proof in order to get an algorithmic proof.
http://hlombardi.free.fr/publis/LectSlides3.pdf
Lesson 4: (elimination of prime cones)
Ordered fields. Real closure. Positivstellensätze.
http://hlombardi.free.fr/publis/LectSlides4.pdf